TDT4195: Visual Computing Fundamentals

The Gaussian

First things first. You're going to see the Gaussian appear all over this course, and especially in the image processing part. You might as well learn it by heart from the get-go. The Gaussian in one dimension:

We can see from this that the Gaussian is separable, yay! This means that we typically apply two one-dimensional gauss filter operations (one in the x direction and one in the y direction) instead of a two-dimensional gauss directly over the entire image.

Graphics

Lab exercises

You still have to figure out how to answer the questions yourself, but the code presented in the tutorials solve the tasks given in the mentioned labs.

Rasterization Algorithms

Rasterisation (or rasterization) is the task of taking an image described in a vector graphics format (shapes) and converting it into a raster image (pixels or dots) for output on a video display or printer, or for storage in a bitmap file format.

Bresenham Line Algorithm

The Bresenham line algorithm is used to determine which points in a raster should be plotted to form a close approximation to a straight line between two points.

In the picture we have a line going from starting point (x0, y0) to end point (x1, y1). The grey squares are the pixels used for drawing the approximation of the line.

In this example we choose the slope to always be between 0 and 1. This means we always increment x, and sometimes increment y. We decide if we should increment y by looking at the error. The error is the distance between the actual point on the line and our current approximation. If the error is greater than 0.5 we increment y by 1, and reduce the error by 1.

Since the slope is always between 0 and 1 in this algorithm it will only work in the first octant. To extend the algorithm to work in every octant we can do as follows:

Circle Rasterization

Circles possess 8–way symmetry, so it is sufficient to calculate one octant and derive the rest.

Bresenham Circle Algorithm

The radius of the circle is r

The center of the circle is pixel (0, 1)

The algorithm starts with pixel (0, r)

It draws a circular arc in the second octant

Coordinate x is incremented at every step

If the value of the circle function becomes non-negative (pixel not inside the circle), y is decremented

Point in Polygon Tests

Draw a ray from pixel p in any direction

Count the number of intersections of the line with the polygon P

If #intersections == odd number then p is inside P

Otherwise p is outside P

Triangle Rasterization Algorithm

A triangle is the simplest polygon shape. To determine the pixels in a triangle, perform an inside test on all pixels of the triangle's bounding box. If the three line functions (from the borders of the triangle) give the same sign for a given pixel, the pixel is inside the triangle.

Area Filling Algorithms

There are multiple ways of filling an area. Flood filling is a simple approach. It starts with a pixel in the area. This pixel is colored. The neighbours are found. The colour-function is called recursively on each of these, checking if they are within the area first.

Anti-aliasing Techniques

Pre-filtering:

extract high frequencies before sampling

treat the pixel as a finite area

compute the % contribution of each primitive in the pixel area

Post-filtering:

extract high frequencies after sampling

increase sampling frequency

results are averaged down

Catmull’s Algorithm

Line clipping

Cohen – Sutherland (CS) Algorithm

Perform a low-cost test which decides if a line segment is entirely inside or entirely outside the clipping window

For each non-trivial line segment compute its intersection with one of the lines defined by the window boundary

Recursively apply the algorithm to both resultant line segments

The low cost test can be done by assigning a 4-bit code to each of the nine sections around a pixel.

Let the endpoints of a line segment be $c1$ and $c2$. Then we can do the following tests.

If $c1 \vee c2 = 0000$

Then the line segment is entirely inside

If $c1 \wedge c2 \neq 0000$

Then the line segment is entirely outside

Skala Algorithm

Liang - Barsky (LB) Algorithm

Yow this shit it based on the parametric equation of the line segment to be clipped.
A line segment is defined by two points.
Name the left-most point $p_1 (x_1, y_1)$ and the other point $p_2 (x_2, y_2)$ so that $x_1 \leq x_2$.
We define $\Delta x = x_2 - x_1$ and $\Delta y = y_2 - y_1$.

The clipping window is a rectangle defined by $x_{\text{min}}$, $x_{\text{max}}$, $y_{\text{min}}$, $y_{\text{max}}$.

A point $q$ on the line segment defined by $p_1$ and $p_2$ is inside the clipping window if it satisfies the following:

If $t_{\text{in}} \leq t_{\text{out}}$ we plug the values into the parametric line equation.
Otherwise there is no intersection with the clipping window.

Polygon clipping

Sutherland - Hodgman (SH) Algorithm

Greiner - Hormann Algorithm

2D and 3D Coordinate Systems and Transformations

A point in euclidian space can be defined as 3D vector. Linear transformation is achieved by post-multiplying the point to a 3x3 matrix.

Affine transformations

Affine transformations are transformations which preserve important geometric properties of the objects being transformed

There are four basic affine transformations:

Translation

Scaling

Rotation

Shearing

Transformation Matrices

Quick note: a hyperplane refers to a substance of one dimensionality less than its ambient space.
I.e. in a 3-dimensional space, the $xy$-plane is a hyperplane. In a 2-dimensional space, the $x$ and $y$ axis are hyperplanes.

Translation

Translation defines movement by a certain distance in a certain direction.
Translation of a point p by a vector $\vec{\textbf{d}}$ results in a new point $\textbf{p'} = \textbf{p} + \vec{\textbf{d}}$.
All we do is add the vector to the point.

The translation transformation matrix is an instantiation of the general affine transformation, $\Phi(p) = A \cdot p + \vec{t}$ with $A = I$ and $\vec{t} = \vec{d}$.

The $n$-dimensional translation matrix is derived by instantiating the general affine transformation with $A = I(n)$.
($I(n)$ is the $n$-dimensional identity matrix)
If you need the homogenous variant, use $I(n+1)$.

The inverse translation of $T(\vec{d})$ is $T^{-1}(\vec{d}) = T(-\vec{d})$.

Scaling

We have $n$ scaling factors $s_{i}$, $i = 1, 2, 3, ..., n$ corresponding to the dimensionality of the space.
If a scaling factor $s_{i} < 1$ the object's size is reduced in the respective dimension, whileas if $s_{i} > 1$ it is increased.
$s_{i} = 1$ has no effect.
$s_{i} = 0$ reduces the object's size in the respective dimension to zero.
If $s_{i} = 0$, $\forall i$ the object disappears.

Mirroring about a major hyperplane can be done by using $s_{i} = -1$ as the factor for the hyperplane which isn't involved in the mirroring.
I.e. to perform 2D mirroring about the $x$ axis, use $S(1, -1)$.
To perform 3D mirroring about the $xy$-plane, use $S(1, 1, -1)$.

For 2D homogenous scaling, remove the third row and third column.
For 3D non-homogenous scaling, remove the fourth row and fourth column.
For 2D non-homogenous scaling, remove the third and fourth rows and columns.

Quaternions

To understand what quaternions are, consider real numbers as the 1D number line and complex numbers as the 2D complex plane. Quaternions are "4D numbers".

In this course we can use quaternions as 4D vectors with the axes $i, j, k, w$.
If you have a $(x, y, z)$ point or vector you can plug these and $w=0$ into the equation $xi + yj + zk + w = 0$ for a quaternion representation.

Rotation

We can use quaternions to rotate points around a vector.
For the calculation we need

$p$, the point that is to be rotated

$\vec{v}$, the vector that $p$ is to be rotated around.

Normalize $\vec{v}$ to the unit vector $\hat{v}$.

$\theta$, the arc of the rotation (how many "degrees" to rotate if you will).

Digital Image Processing

Typical image processing steps

Image aquisition

Image enhancement

Image restoration

Morphological processing

Segmentation

Representation and description

Object recognition

The Human Eye

The human eye consists of the pupil, the lens and the retina. Light enters through the pupil. The lens focuses light onto the retina. The retina consists of nerve cells called photoreceptors. There are types of receptors, cones and rods. A typical eye has 6-7 million cones, each connected to a dedicated nerve end. Cones enable color vision. It also has 75-150 million rods, several connected to one nerve end. Rods allows logarithmic light sensitivity.

Sampling and Quantization

Sampling and quantization used when converting a stream of continuous data into digital form. Formalized: A continuous function $f(t)$ is to be sampled every $T$ steps of $t$ (which typically represents time). We say that $T$ is the sampling interval. The sampled function is then the sequence of values $f_n = f(nT), n \in \mathbb{N}$. Images are typically represented as digitized streams of two dimenstions, x and y.

Sampling Theorem

When a stream contains higher frequencies than the sampling frequency can handle, unwanted artifacts known as aliasing are produced. The Nyquist-Shannon sampling theorem formalizes this: $$f_s \geq 2 \cdot f_{max}$$

This implies that sampling should be performed with a frequency twice as large as the highest frequency that occurs in the signal to avoid aliasing.

Image enhancement

Image enhancement typically aims to do things like: noise removal, highlight interesting details, make the image more visually appealing. There are two main categories of techniques: spatial domain techniques, and transform domain techniques.

Histograms

Histograms of an image provides information about the distribution of intensity levels of an image. Both global (entire image) and local (parts of the image) histograms are useful.

Histogram Equalization

Compute the gray value histogram of the image (I):

$$h(k) = \sum \limits_{ij \in \Omega} \delta (I_{ij} - k)$$

Compute the cummulative proportion of pixels with a gray-value smaller than i:

Spatial domain enhancement techniques

Involves direct manipulation of pixels, with or without considering neighboring pixels. Spatial image enhancement techniques that do not consider a pixel's neighborhood are called intensity transformations or point processing operations. Intensity transformations change the value of each pixel based on its intensity alone. Examples include: image negatives, contrast stretching, gamma transform, thesholding/binarization.

Neighborhood

Spatial Filtering

A spatial filter exists of a neighborhood, associated weights for each pixel in the neigborhood, and a predefined operation on the weighted pixels. When the weights sum to 1, the gray value is not changed.

Smoothing

We can make an averaging spatial filter to smooth an image. Consider an square 8-neighborhood, and the following weights:

This results in a smoother version of the image, which reduces noise. This averging filter is also known as the box-filter.

The averaging spatial filter is a linear filter. An example of a popular non-linear smoothing filter is the median filter. The median filter sets a pixel to median of itself and its neighbors.

Convolution

Convolution is written as $(f*g)(t)$ meaning the function $f$ convolutes $g$ with variable $x$.
Convolution can be understood as one wave, or function, travelling through another as shown in the illustration below.
The resulting value (the convolution of f through g) is the area covered by both graphs.
Now look at the image and notice that

The convolution $(f*g)(t)$ is zero when the waves are not overlapping.

The convolution $(f*g)(t)$ grows when f is moving through g, before reaching the edge of g.

The convolution $(f*g)(t)$ is at is maximum when the intersection of the waves is biggest.

The convolution $(f*g)(t)$ shrinks when the front of f has reached the end of g and has begun "leaving" it.

Remember that we are looking for the area, unlike when colliding water waves where two large wave tops will give us a twice the height.

Sharpening

We use Laplace for this. TODO: write about this.

Transform domain enhancement techniques

Involves transforming the image into a different representation. Examples of transforms include fourier transforms and wavelet transforms.

Frequency domain

Filtering can be done in the frequency domain. We use the discrete fourier transform to enable this. The Discrete Fourier-Transform (DFT) is defined as:

Morphology

Morphology is a set of image processing operations based on shapes. This means adding or removing pixels on the boundaries of objects. It is done by taking an image, performing an operation on each pixel with the use of a structuring element and creating an output image of the same size. A structuring element is a shape used to define the neighbourhood of the pixels of interest.

Morphological operations

Dilation and erosion are the most basic morphological operations that when combined make up the opening and closing operations.

Dilation

Dilation adds pixels to an image. This is done by applying the appropriate rule to the pixels of the neighbourhood and assign a value to the corresponding pixel in the output image. The picture below is illustrates the dilution of a binary image. In the figure, the morphological dilation function sets the value of the output pixel to 1 because 1 is the highest value in the neighbourhood defined by the structuring element. Pixels beyond the image border are assigned the minimum value afforded by the data type.

The following figure illustrates dilation for a grayscale image. Note how the function looks at all the pixels in the neighborhood and uses the highest value of all as the corresponding pixel in the output image.

The notation for dilation is $f \oplus s$, where $f$ is the image and $s$ is the structuring element

Erosion

Erosion removes pixels from an image. This is done the same way as can be seen in the binary dilation figure seen above, except the function looks for 0 instead of 1. Pixels beyond the image border are assigned the maximum value afforded by the data type.

The notation for erosion is $f \ominus s$, where $f$ is the image and $s$ is the structuring element.

The following image shows an eroded image compared to the original image. Notice how there are less white pixels (1s) in the eroded picture.